STAT 330
EXTRA PROBLEMS CHAPTERS 1 TO 7
1. Consider the following functions:
(a) f (x) = kx(0.3)x ,
x = 1, 2, 3, .
2 1
(b) f (x) = k (1 + x )
|x|
(c) f (x) = ke
< x <
,
2 x
(d) f (x) = kx e
,
( +1)
(e) f (x) = kx
< x <
,
,
x > 0, > 0
x > > 0, > 0
In e
Midterm 2 Review
1. Suppose X U N IF (0, 1) and Y U N IF (0, 1) independently. Find the joint p.d.f. of
U = (2 log X )1/2 cos(2Y )
V = (2 log X )1/2 sin(2Y ).
Answer:
g (u, v ) =
1 1 (u2 +v2 )
e2
, <u<, <v <
2
2. Let X N (0, 1) and Y N (0, 1) independentl
Solutions to Practice Questions
1.
(a) W1 N (0, 4 2 ) since linear combinations of independent normal r.v.s are also normal.
X1 X2
2 2
Y2 = X3 X4
2 2
(b) Note that
and
N (0, 1) and
X3 X4
2 2
N (0, 1). Therefore, Y1 =
X1 X2
2 2
2 1)
(
2 1). Furthermore,
Stat 330 - Tutorial 1
1. Suppose X N (, 2 ).
(a) Show that is a location parameter.
(b) Show that if = 0, then is a scale parameter.
2. If U is uniformly distributed on (0, 1), nd the probability density function of
3. If U is uniformly distributed on (1,
Stat 330 - Tutorial 1
1.
( x ) 2
1
.
(a) X N (, 2 ) = f (x; ) = exp
2 2
2
1
x2
Note that f0 (x) = f (x; = 0) = exp 2 .
2
2
Since f0 (x ) = f (x; ), by denition, is a location parameter.
1
x2
(b) If = 0, f (x; ) = exp 2 .
2
2
1
x2
Note that f1 (x) = f
Stat 330 - Tutorial 2
1. Suppose X BIN (n, p).
(a) Find E (X (k) ) and use the result to nd E (X ) and V ar(X ).
(b) Find the mgf of X and use the mgf to nd E (X ) and V ar(X ).
2. 3.2.5 (Page 31 of supplementary notes).
The Hardy-Weinberg law of genetics
Stat 330 - Tutorial 3
Exercise 3.3.6 (Page 34 of supplementary notes). Suppose X and Y are continuous random
variables with joint p.d.f.
f (x, y ) =
k
, 0 < x < , 0 < y <
(1 + x + y )3
and 0 otherwise. Determine k . Find
1. P X 1 , Y
3
1
2
2. P (X Y )
3
University of Waterloo
STAT 330 Term Test #2
Term: Spring
Year:
2012
Student Name (Print):
UW Student ID Number:
UW Student Userid:
Instructor (Circle one):
1. Javid Ali
2. Pengfei Li
Course Abbreviation and Number:
Course Title:
Stat 330
Mathematical Sta
Stat 330 - Tutorial 5
1. (Minimum Mean Squared Error) Let h(X ) be any function of X and g (X ) = E (Y |X ).
Show that the g (X ) minimizes the mean squared error, that is,
E [(Y g (X )2 ] E [(Y h(X )2 ].
Hint: Write E [(Y h(X )2 ] = E [(A + B )2 ], where
Stat 330 - Tutorial 7
1. (Question 28 from the back of the course notes) Assume that Y denotes the number
of bacteria in a cubic centimetre of liquid and that Y | = P OI (). Further assume that varies from location to location and the corresponding random
Stat 330 - Tutorial 9
iid
1. Suppose X1 , . . . , Xn N (, 2 ) where is known. Suppose we would like to test H0 :
= 0 vs HA : = 0 .
(a) Construct the LR test statistic under H0 .
(b) Show that the 2 distribution is the exact distribution of the test stati
Stat 330 - Tutorial 7 Solution
1.
t 1)
Y | = P oi() = E (Y | = ) = , V ar(Y | = ) = , MY |= (t) = e(e
= E (Y |) = , V ar(Y |) = , MY | (t) = e(e
t 1)
Gamma(, ) = E () = , V ar() = 2 , M (t) = (1 t)
Therefore,
E (Y ) = E [E (Y |)] = E [] =
V ar(Y ) = E [
Stat 330 - Tutorial 5 Solution
1.
E [(Y h(X )2 ] = E
Y g (X ) + g (X ) h(X )
=E
Y g (X )
2
=E
Y g (X )
2
2
+ 2 Y g (X ) g (X ) h(X ) + g (X ) h(X )
+ 2E
Y g (X ) g (X ) h(X )
+E
2
g (X ) h(X )
0 (expectation
of squared terms)
E
Y g (X )
2
+ 2E
Y g (X ) g
Stat 330 - Tutorial 9 Solution
Brief solutions are available below.
1.
(a) We can write
l() = log(C )
1
2 2
n
(xi )2
i=1
where C is some constant, not in terms of . Therefore, the estimator is = X .
You can show that the LR test statistic under H0 : = 0
Tutorial 3 Solution
We are given
f (x, y ) =
k
, 0 < x < , 0 < y <
(1 + x + y )3
and 0 otherwise.
The integral over the entire rst quadrant (the support of (X, Y ) should be equal to 1. That is,
0
0
=
k
k
dxdy = 1 =
dy = 1
(1 + x + y )3
2(1 + x + y )2 x=
University of Waterloo
STAT 330 Term Test #1
Term: Spring
Year:
2012
Student Name (Print):
UW Student ID Number:
UW Student Userid:
Instructor (Circle one):
1. Javid Ali
2. Pengfei Li
Course Abbreviation and Number:
Course Title:
Stat 330
Mathematical Sta
Midterm 1 Review
1. Suppose f (x, y ) = kxey , 0 < x < 1, 0 < y < and f (x, y ) = 0 otherwise.
(a) Show that k = 2.
Solution:
1
0
1
k=1
2
=
f (x, y )dydx = 1
0
=
k = 2.
(b) Are X and Y independent?
Solution: X and Y are independent since
the support is r
STAT 330
Mathematical Statistics
Term Test 1 Solution
Problem I: (6pts) If X has a Beta distribution with parameters a and b, then the p.d.f. is
f (x) =
(a + b) a1
x (1 x)b1 , 0 < x < 1, a > 0, b > 0
(a)(b)
and 0 otherwise.
(a) (3pts) Find E (X ) with a =
Assignment #2 (Due 11/5)
Note: Please submit the assignment to Drop Box 15, slot 2 (Y. Ning) and slot 4
(P. Balka) outside MC 4006/4007. The assignment will be collected at 5:00pm.
Late submissions will not be marked and will be assessed a grade of zero !
STAT 330 - Assignment #4
Due Monday, Dec. 2 at 5:00 pm in Drop Box 15 (slot 2 (Y. Ning) and slot 4 (P. Balka) outside MC
4006/4007.
1) Let X i ~ U ( 0 ,1), i 1, 2 ,., n independen tly
a) Find the limiting distribution of Y n nX
(where X (1 ) min (X 1 , X
STAT 330 F13 Final Exam Information
Time and Location: See official exam schedule.
Format:
Same format and style as term tests.
Exam aides: None. NO CALCULATORS. Relevant distributions (e.g. chi-square, gamma,.) will be
provided if and where necessary. Yo
Assignment #1 (Due Oct 10th)
Note: Please submit the assignment to Drop Box 15, slot 2 (Y. Ning) and slot 4
(P. Balka) outside MC 4006/4007. The assignment will be collected at 5:00pm.
Late submissions will not be marked and will be assessed a grade of ze
STAT 330 F13 Term Test I Information
Time and Date: Friday, Oct.18. 3:30 4:20.
Location:
Section 001 (Y. Ning):
A P (Last Name): MC 2017
QZ
: MC 2054
Section 002 (P. Balka):
DC 1350
Format:
Closed book. Short answer (3 or 4 questions). The same format and
STAT 330 F13 Term Test II Information
Time and Date: Friday, Nov. 22, 3:30 4:20.
Location: Same as for Term Test I:
Section 001 (Y. Ning):
A P (Last Name): MC 2017
QZ
: MC 2054
Section 002 (P. Balka):
DC 1350
Format:
Same format and style as Term Test I
E
STAT 330
Mathematical Statistics
Assignment 1
Due: Thursday, May 31, 2012
1. Suppose X follows Geo(p) distribution with 0 < p < 1. That is,
f (x) = (1 p)x p, x = 0, 1, . . .
with 0 < p < 1.
(a) Find the mgf of X , M (t), and indicate when M (t) exists.
(b
STAT 330
Mathematical Statistics
Assignment 1 Solution
1. (a) Using the identity for an innite geometric series,
tx
x
[(1 p)et ]x p
e (1 p) p =
M (t) =
x=0
x=0
=
p
assuming (1 p)et < 1 = t < log(1 p)
1 (1 p)et
since the common ratio must be less than 1 fo
STAT 330
Mathematical Statistics
Assignment 2
Due: Thursday, June 28, 2012 in class
1. If X is a non-negative continuous random variable, show that
[1 F (x)]dx
E (X ) =
0
Hint: Consider the right hand side. Write F (x) =
order of integration.
x
0
fX (t)dt
STAT 330
Mathematical Statistics
1. Since F (x) =
x
0
Assignment 2 Solution
fX (x)dx,
[1 F (x)]dx =
0
x
1
0
fX (t)dt dx =
0
fX (t)dtdx
0
x
fX (t)dtdx
=
x
0
The region of integration is shown in red below. Switching the order of integration, we have
t
[1 F
STAT 330
Mathematical Statistics
Assignment 3
Due: Thursday, July 26, 2012 by 5pm in M3 3112
1. Suppose X1 , X2 , . . . , Xn are i.i.d. random variables with E (Xi ) = , V ar(Xi ) =
2 < and E (Xi4 ) = 4 < . Let
1
Xn =
n
n
2
Sn
and
Xi
i=1
1
=
n1
n
(Xi Xn